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Theorem dchrisum0lem1 27578

Description: Lemma for dchrisum0 27582. (Contributed by Mario Carneiro, 12-May-2016.) (Revised by Mario Carneiro, 7-Jun-2016.)

**Hypotheses**
Ref	Expression
rpvmasum.z	⊢ 𝑍 = (ℤ/nℤ‘𝑁)
rpvmasum.l	⊢ 𝐿 = (ℤRHom‘𝑍)
rpvmasum.a	⊢ (𝜑 → 𝑁 ∈ ℕ)
rpvmasum2.g	⊢ 𝐺 = (DChr‘𝑁)
rpvmasum2.d	⊢ 𝐷 = (Base‘𝐺)
rpvmasum2.1	⊢ 1 = (0_g‘𝐺)
rpvmasum2.w	⊢ 𝑊 = {𝑦 ∈ (𝐷 ∖ { 1 }) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}
dchrisum0.b	⊢ (𝜑 → 𝑋 ∈ 𝑊)
dchrisum0lem1.f	⊢ 𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / (√‘𝑎)))
dchrisum0.c	⊢ (𝜑 → 𝐶 ∈ (0[,)+∞))
dchrisum0.s	⊢ (𝜑 → seq1( + , 𝐹) ⇝ 𝑆)
dchrisum0.1	⊢ (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑆)) ≤ (𝐶 / (√‘𝑦)))

**Assertion**
Ref	Expression
dchrisum0lem1	⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))) ∈ 𝑂(1))

Distinct variable groups: 𝑥,𝑚,𝑦, 1 𝑚,𝑑,𝑥,𝑦,𝐶 𝐹,𝑑,𝑥,𝑦 𝑎,𝑑,𝑚,𝑥,𝑦 𝑚,𝑁,𝑥,𝑦 𝜑,𝑑,𝑚,𝑥 𝑆,𝑑,𝑚,𝑥,𝑦 𝑥,𝑊 𝑚,𝑍,𝑥,𝑦 𝐷,𝑚,𝑥,𝑦 𝐿,𝑎,𝑑,𝑚,𝑥,𝑦 𝑋,𝑎,𝑑,𝑚,𝑥,𝑦 𝑚,𝐹 Allowed substitution hints: 𝜑(𝑦,𝑎) 𝐶(𝑎) 𝐷(𝑎,𝑑) 𝑆(𝑎) 1 (𝑎,𝑑) 𝐹(𝑎) 𝐺(𝑥,𝑦,𝑚,𝑎,𝑑) 𝑁(𝑎,𝑑) 𝑊(𝑦,𝑚,𝑎,𝑑) 𝑍(𝑎,𝑑)

**Proof of Theorem dchrisum0lem1**
Step	Hyp	Ref	Expression
1		fzfid 14024	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1...(⌊‘𝑥)) ∈ Fin)
2		fzfid 14024	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2))) ∈ Fin)
3		fzfid 14024	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑))) ∈ Fin)
4		elfznn 13613	. . . . . . 7 ⊢ (𝑑 ∈ (1...(⌊‘𝑥)) → 𝑑 ∈ ℕ)
5		elfzuz 13580	. . . . . . 7 ⊢ (𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑))) → 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))
6	4, 5	anim12i 612	. . . . . 6 ⊢ ((𝑑 ∈ (1...(⌊‘𝑥)) ∧ 𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))) → (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1))))
7	6	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((𝑑 ∈ (1...(⌊‘𝑥)) ∧ 𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))) → (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))))
8		elfzuz 13580	. . . . . . 7 ⊢ (𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2))) → 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))
9		elfznn 13613	. . . . . . 7 ⊢ (𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚))) → 𝑑 ∈ ℕ)
10	8, 9	anim12ci 613	. . . . . 6 ⊢ ((𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1))))
11	10	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))))
12		eluzelz 12913	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)) → 𝑚 ∈ ℤ)
13	12	ad2antll 728	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → 𝑚 ∈ ℤ)
14	13	zred 12747	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → 𝑚 ∈ ℝ)
15		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ⁺)
16		2z 12675	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℤ
17		rpexpcl 14131	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 2 ∈ ℤ) → (𝑥↑2) ∈ ℝ⁺)
18	15, 16, 17	sylancl 585	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑2) ∈ ℝ⁺)
19	18	rpred 13099	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑2) ∈ ℝ)
20	19	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑥↑2) ∈ ℝ)
21		simprl 770	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → 𝑑 ∈ ℕ)
22	21	nnrpd 13097	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → 𝑑 ∈ ℝ⁺)
23	14, 20, 22	lemuldivd 13148	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → ((𝑚 · 𝑑) ≤ (𝑥↑2) ↔ 𝑚 ≤ ((𝑥↑2) / 𝑑)))
24	21	nnred 12308	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → 𝑑 ∈ ℝ)
25	15	rprege0d 13106	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
26		flge0nn0 13871	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ₀)
27		nn0p1nn 12592	. . . . . . . . . . . . . 14 ⊢ ((⌊‘𝑥) ∈ ℕ₀ → ((⌊‘𝑥) + 1) ∈ ℕ)
28	25, 26, 27	3syl 18	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((⌊‘𝑥) + 1) ∈ ℕ)
29	28	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → ((⌊‘𝑥) + 1) ∈ ℕ)
30		simprr 772	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))
31		eluznn 12983	. . . . . . . . . . . 12 ⊢ ((((⌊‘𝑥) + 1) ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1))) → 𝑚 ∈ ℕ)
32	29, 30, 31	syl2anc 583	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → 𝑚 ∈ ℕ)
33	32	nnrpd 13097	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → 𝑚 ∈ ℝ⁺)
34	24, 20, 33	lemuldiv2d 13149	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → ((𝑚 · 𝑑) ≤ (𝑥↑2) ↔ 𝑑 ≤ ((𝑥↑2) / 𝑚)))
35	23, 34	bitr3d 281	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑚 ≤ ((𝑥↑2) / 𝑑) ↔ 𝑑 ≤ ((𝑥↑2) / 𝑚)))
36		rpcn 13067	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℂ)
37	36	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℂ)
38	37	sqvald 14193	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑2) = (𝑥 · 𝑥))
39	38	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑥↑2) = (𝑥 · 𝑥))
40		simplr 768	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → 𝑥 ∈ ℝ⁺)
41	40	rpred 13099	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → 𝑥 ∈ ℝ)
42		reflcl 13847	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ → (⌊‘𝑥) ∈ ℝ)
43		peano2re 11463	. . . . . . . . . . . . . . . 16 ⊢ ((⌊‘𝑥) ∈ ℝ → ((⌊‘𝑥) + 1) ∈ ℝ)
44	41, 42, 43	3syl 18	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → ((⌊‘𝑥) + 1) ∈ ℝ)
45		fllep1 13852	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ → 𝑥 ≤ ((⌊‘𝑥) + 1))
46	41, 45	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → 𝑥 ≤ ((⌊‘𝑥) + 1))
47		eluzle 12916	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)) → ((⌊‘𝑥) + 1) ≤ 𝑚)
48	47	ad2antll 728	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → ((⌊‘𝑥) + 1) ≤ 𝑚)
49	41, 44, 14, 46, 48	letrd 11447	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → 𝑥 ≤ 𝑚)
50	41, 14, 40	lemul1d 13142	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑥 ≤ 𝑚 ↔ (𝑥 · 𝑥) ≤ (𝑚 · 𝑥)))
51	49, 50	mpbid 232	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑥 · 𝑥) ≤ (𝑚 · 𝑥))
52	39, 51	eqbrtrd 5188	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑥↑2) ≤ (𝑚 · 𝑥))
53	20, 41, 33	ledivmuld 13152	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (((𝑥↑2) / 𝑚) ≤ 𝑥 ↔ (𝑥↑2) ≤ (𝑚 · 𝑥)))
54	52, 53	mpbird 257	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → ((𝑥↑2) / 𝑚) ≤ 𝑥)
55		nnre 12300	. . . . . . . . . . . . 13 ⊢ (𝑑 ∈ ℕ → 𝑑 ∈ ℝ)
56	55	ad2antrl 727	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → 𝑑 ∈ ℝ)
57	20, 32	nndivred 12347	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → ((𝑥↑2) / 𝑚) ∈ ℝ)
58		letr 11384	. . . . . . . . . . . 12 ⊢ ((𝑑 ∈ ℝ ∧ ((𝑥↑2) / 𝑚) ∈ ℝ ∧ 𝑥 ∈ ℝ) → ((𝑑 ≤ ((𝑥↑2) / 𝑚) ∧ ((𝑥↑2) / 𝑚) ≤ 𝑥) → 𝑑 ≤ 𝑥))
59	56, 57, 41, 58	syl3anc 1371	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → ((𝑑 ≤ ((𝑥↑2) / 𝑚) ∧ ((𝑥↑2) / 𝑚) ≤ 𝑥) → 𝑑 ≤ 𝑥))
60	54, 59	mpan2d 693	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑑 ≤ ((𝑥↑2) / 𝑚) → 𝑑 ≤ 𝑥))
61	35, 60	sylbid 240	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑚 ≤ ((𝑥↑2) / 𝑑) → 𝑑 ≤ 𝑥))
62	61	pm4.71rd 562	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑚 ≤ ((𝑥↑2) / 𝑑) ↔ (𝑑 ≤ 𝑥 ∧ 𝑚 ≤ ((𝑥↑2) / 𝑑))))
63		nnge1 12321	. . . . . . . . . . . . . 14 ⊢ (𝑑 ∈ ℕ → 1 ≤ 𝑑)
64	63	ad2antrl 727	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → 1 ≤ 𝑑)
65		1re 11290	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℝ
66		0lt1 11812	. . . . . . . . . . . . . . 15 ⊢ 0 < 1
67	65, 66	pm3.2i 470	. . . . . . . . . . . . . 14 ⊢ (1 ∈ ℝ ∧ 0 < 1)
68	22	rpregt0d 13105	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑑 ∈ ℝ ∧ 0 < 𝑑))
69	18	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑥↑2) ∈ ℝ⁺)
70	69	rpregt0d 13105	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → ((𝑥↑2) ∈ ℝ ∧ 0 < (𝑥↑2)))
71		lediv2 12185	. . . . . . . . . . . . . 14 ⊢ (((1 ∈ ℝ ∧ 0 < 1) ∧ (𝑑 ∈ ℝ ∧ 0 < 𝑑) ∧ ((𝑥↑2) ∈ ℝ ∧ 0 < (𝑥↑2))) → (1 ≤ 𝑑 ↔ ((𝑥↑2) / 𝑑) ≤ ((𝑥↑2) / 1)))
72	67, 68, 70, 71	mp3an2i 1466	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (1 ≤ 𝑑 ↔ ((𝑥↑2) / 𝑑) ≤ ((𝑥↑2) / 1)))
73	64, 72	mpbid 232	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → ((𝑥↑2) / 𝑑) ≤ ((𝑥↑2) / 1))
74	20	recnd 11318	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑥↑2) ∈ ℂ)
75	74	div1d 12062	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → ((𝑥↑2) / 1) = (𝑥↑2))
76	73, 75	breqtrd 5192	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → ((𝑥↑2) / 𝑑) ≤ (𝑥↑2))
77		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1))) → 𝑑 ∈ ℕ)
78		nndivre 12334	. . . . . . . . . . . . 13 ⊢ (((𝑥↑2) ∈ ℝ ∧ 𝑑 ∈ ℕ) → ((𝑥↑2) / 𝑑) ∈ ℝ)
79	19, 77, 78	syl2an 595	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → ((𝑥↑2) / 𝑑) ∈ ℝ)
80		letr 11384	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ ℝ ∧ ((𝑥↑2) / 𝑑) ∈ ℝ ∧ (𝑥↑2) ∈ ℝ) → ((𝑚 ≤ ((𝑥↑2) / 𝑑) ∧ ((𝑥↑2) / 𝑑) ≤ (𝑥↑2)) → 𝑚 ≤ (𝑥↑2)))
81	14, 79, 20, 80	syl3anc 1371	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → ((𝑚 ≤ ((𝑥↑2) / 𝑑) ∧ ((𝑥↑2) / 𝑑) ≤ (𝑥↑2)) → 𝑚 ≤ (𝑥↑2)))
82	76, 81	mpan2d 693	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑚 ≤ ((𝑥↑2) / 𝑑) → 𝑚 ≤ (𝑥↑2)))
83	35, 82	sylbird 260	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑑 ≤ ((𝑥↑2) / 𝑚) → 𝑚 ≤ (𝑥↑2)))
84	83	pm4.71rd 562	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑑 ≤ ((𝑥↑2) / 𝑚) ↔ (𝑚 ≤ (𝑥↑2) ∧ 𝑑 ≤ ((𝑥↑2) / 𝑚))))
85	35, 62, 84	3bitr3d 309	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → ((𝑑 ≤ 𝑥 ∧ 𝑚 ≤ ((𝑥↑2) / 𝑑)) ↔ (𝑚 ≤ (𝑥↑2) ∧ 𝑑 ≤ ((𝑥↑2) / 𝑚))))
86		fznnfl 13913	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ → (𝑑 ∈ (1...(⌊‘𝑥)) ↔ (𝑑 ∈ ℕ ∧ 𝑑 ≤ 𝑥)))
87	86	baibd 539	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ ∧ 𝑑 ∈ ℕ) → (𝑑 ∈ (1...(⌊‘𝑥)) ↔ 𝑑 ≤ 𝑥))
88	41, 21, 87	syl2anc 583	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑑 ∈ (1...(⌊‘𝑥)) ↔ 𝑑 ≤ 𝑥))
89	79	flcld 13849	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (⌊‘((𝑥↑2) / 𝑑)) ∈ ℤ)
90		elfz5 13576	. . . . . . . . . 10 ⊢ ((𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)) ∧ (⌊‘((𝑥↑2) / 𝑑)) ∈ ℤ) → (𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑))) ↔ 𝑚 ≤ (⌊‘((𝑥↑2) / 𝑑))))
91	30, 89, 90	syl2anc 583	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑))) ↔ 𝑚 ≤ (⌊‘((𝑥↑2) / 𝑑))))
92		flge 13856	. . . . . . . . . 10 ⊢ ((((𝑥↑2) / 𝑑) ∈ ℝ ∧ 𝑚 ∈ ℤ) → (𝑚 ≤ ((𝑥↑2) / 𝑑) ↔ 𝑚 ≤ (⌊‘((𝑥↑2) / 𝑑))))
93	79, 13, 92	syl2anc 583	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑚 ≤ ((𝑥↑2) / 𝑑) ↔ 𝑚 ≤ (⌊‘((𝑥↑2) / 𝑑))))
94	91, 93	bitr4d 282	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑))) ↔ 𝑚 ≤ ((𝑥↑2) / 𝑑)))
95	88, 94	anbi12d 631	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → ((𝑑 ∈ (1...(⌊‘𝑥)) ∧ 𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))) ↔ (𝑑 ≤ 𝑥 ∧ 𝑚 ≤ ((𝑥↑2) / 𝑑))))
96	20	flcld 13849	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (⌊‘(𝑥↑2)) ∈ ℤ)
97		elfz5 13576	. . . . . . . . . 10 ⊢ ((𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)) ∧ (⌊‘(𝑥↑2)) ∈ ℤ) → (𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2))) ↔ 𝑚 ≤ (⌊‘(𝑥↑2))))
98	30, 96, 97	syl2anc 583	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2))) ↔ 𝑚 ≤ (⌊‘(𝑥↑2))))
99		flge 13856	. . . . . . . . . 10 ⊢ (((𝑥↑2) ∈ ℝ ∧ 𝑚 ∈ ℤ) → (𝑚 ≤ (𝑥↑2) ↔ 𝑚 ≤ (⌊‘(𝑥↑2))))
100	20, 13, 99	syl2anc 583	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑚 ≤ (𝑥↑2) ↔ 𝑚 ≤ (⌊‘(𝑥↑2))))
101	98, 100	bitr4d 282	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2))) ↔ 𝑚 ≤ (𝑥↑2)))
102		fznnfl 13913	. . . . . . . . . 10 ⊢ (((𝑥↑2) / 𝑚) ∈ ℝ → (𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚))) ↔ (𝑑 ∈ ℕ ∧ 𝑑 ≤ ((𝑥↑2) / 𝑚))))
103	102	baibd 539	. . . . . . . . 9 ⊢ ((((𝑥↑2) / 𝑚) ∈ ℝ ∧ 𝑑 ∈ ℕ) → (𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚))) ↔ 𝑑 ≤ ((𝑥↑2) / 𝑚)))
104	57, 21, 103	syl2anc 583	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚))) ↔ 𝑑 ≤ ((𝑥↑2) / 𝑚)))
105	101, 104	anbi12d 631	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → ((𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) ↔ (𝑚 ≤ (𝑥↑2) ∧ 𝑑 ≤ ((𝑥↑2) / 𝑚))))
106	85, 95, 105	3bitr4d 311	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → ((𝑑 ∈ (1...(⌊‘𝑥)) ∧ 𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))) ↔ (𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚))))))
107	106	ex 412	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1))) → ((𝑑 ∈ (1...(⌊‘𝑥)) ∧ 𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))) ↔ (𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))))))
108	7, 11, 107	pm5.21ndd 379	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((𝑑 ∈ (1...(⌊‘𝑥)) ∧ 𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))) ↔ (𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚))))))
109		ssun2 4202	. . . . . . . 8 ⊢ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑))) ⊆ ((1...(⌊‘𝑥)) ∪ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑))))
110	28	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((⌊‘𝑥) + 1) ∈ ℕ)
111		nnuz 12946	. . . . . . . . . 10 ⊢ ℕ = (ℤ_≥‘1)
112	110, 111	eleqtrdi 2854	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((⌊‘𝑥) + 1) ∈ (ℤ_≥‘1))
113		dchrisum0lem1a 27548	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥 ≤ ((𝑥↑2) / 𝑑) ∧ (⌊‘((𝑥↑2) / 𝑑)) ∈ (ℤ_≥‘(⌊‘𝑥))))
114	113	simprd 495	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (⌊‘((𝑥↑2) / 𝑑)) ∈ (ℤ_≥‘(⌊‘𝑥)))
115		fzsplit2 13609	. . . . . . . . 9 ⊢ ((((⌊‘𝑥) + 1) ∈ (ℤ_≥‘1) ∧ (⌊‘((𝑥↑2) / 𝑑)) ∈ (ℤ_≥‘(⌊‘𝑥))) → (1...(⌊‘((𝑥↑2) / 𝑑))) = ((1...(⌊‘𝑥)) ∪ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))))
116	112, 114, 115	syl2anc 583	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (1...(⌊‘((𝑥↑2) / 𝑑))) = ((1...(⌊‘𝑥)) ∪ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))))
117	109, 116	sseqtrrid 4062	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑))) ⊆ (1...(⌊‘((𝑥↑2) / 𝑑))))
118	117	sselda 4008	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))) → 𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑))))
119		rpvmasum2.g	. . . . . . . . 9 ⊢ 𝐺 = (DChr‘𝑁)
120		rpvmasum.z	. . . . . . . . 9 ⊢ 𝑍 = (ℤ/nℤ‘𝑁)
121		rpvmasum2.d	. . . . . . . . 9 ⊢ 𝐷 = (Base‘𝐺)
122		rpvmasum.l	. . . . . . . . 9 ⊢ 𝐿 = (ℤRHom‘𝑍)
123		rpvmasum2.w	. . . . . . . . . . . . 13 ⊢ 𝑊 = {𝑦 ∈ (𝐷 ∖ { 1 }) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}
124	123	ssrab3 4105	. . . . . . . . . . . 12 ⊢ 𝑊 ⊆ (𝐷 ∖ { 1 })
125		dchrisum0.b	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ∈ 𝑊)
126	124, 125	sselid 4006	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ (𝐷 ∖ { 1 }))
127	126	eldifad 3988	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
128	127	ad3antrrr 729	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑)))) → 𝑋 ∈ 𝐷)
129		elfzelz 13584	. . . . . . . . . 10 ⊢ (𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑))) → 𝑚 ∈ ℤ)
130	129	adantl 481	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑)))) → 𝑚 ∈ ℤ)
131	119, 120, 121, 122, 128, 130	dchrzrhcl 27307	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑)))) → (𝑋‘(𝐿‘𝑚)) ∈ ℂ)
132		elfznn 13613	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑))) → 𝑚 ∈ ℕ)
133	132	adantl 481	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑)))) → 𝑚 ∈ ℕ)
134	133	nnrpd 13097	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑)))) → 𝑚 ∈ ℝ⁺)
135	134	rpsqrtcld 15460	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑)))) → (√‘𝑚) ∈ ℝ⁺)
136	135	rpcnd 13101	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑)))) → (√‘𝑚) ∈ ℂ)
137	135	rpne0d 13104	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑)))) → (√‘𝑚) ≠ 0)
138	131, 136, 137	divcld 12070	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑)))) → ((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) ∈ ℂ)
139	4	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ∈ ℕ)
140	139	nnrpd 13097	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ∈ ℝ⁺)
141	140	rpsqrtcld 15460	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (√‘𝑑) ∈ ℝ⁺)
142	141	rpcnne0d 13108	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((√‘𝑑) ∈ ℂ ∧ (√‘𝑑) ≠ 0))
143	142	adantr 480	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑)))) → ((√‘𝑑) ∈ ℂ ∧ (√‘𝑑) ≠ 0))
144	143	simpld 494	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑)))) → (√‘𝑑) ∈ ℂ)
145	143	simprd 495	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑)))) → (√‘𝑑) ≠ 0)
146	138, 144, 145	divcld 12070	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑)))) → (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) ∈ ℂ)
147	118, 146	syldan 590	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))) → (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) ∈ ℂ)
148	147	anasss 466	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ (1...(⌊‘𝑥)) ∧ 𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑))))) → (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) ∈ ℂ)
149	1, 2, 3, 108, 148	fsumcom2 15822	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑑 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) = Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)))
150	149	mpteq2dva 5266	. 2 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ Σ𝑑 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))) = (𝑥 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))))
151	65	a1i 11	. . 3 ⊢ (𝜑 → 1 ∈ ℝ)
152		2cn 12368	. . . . . . . 8 ⊢ 2 ∈ ℂ
153	15	rpsqrtcld 15460	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (√‘𝑥) ∈ ℝ⁺)
154	153	rpcnd 13101	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (√‘𝑥) ∈ ℂ)
155		mulcl 11268	. . . . . . . 8 ⊢ ((2 ∈ ℂ ∧ (√‘𝑥) ∈ ℂ) → (2 · (√‘𝑥)) ∈ ℂ)
156	152, 154, 155	sylancr 586	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (2 · (√‘𝑥)) ∈ ℂ)
157	141	rprecred 13110	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (1 / (√‘𝑑)) ∈ ℝ)
158	1, 157	fsumrecl 15782	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) ∈ ℝ)
159	158	recnd 11318	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) ∈ ℂ)
160	159, 156	subcld 11647	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) − (2 · (√‘𝑥))) ∈ ℂ)
161		2re 12367	. . . . . . . . . . 11 ⊢ 2 ∈ ℝ
162		dchrisum0.c	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ∈ (0[,)+∞))
163		elrege0 13514	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ (0[,)+∞) ↔ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶))
164	162, 163	sylib 218	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶))
165	164	simpld 494	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ∈ ℝ)
166		remulcl 11269	. . . . . . . . . . 11 ⊢ ((2 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (2 · 𝐶) ∈ ℝ)
167	161, 165, 166	sylancr 586	. . . . . . . . . 10 ⊢ (𝜑 → (2 · 𝐶) ∈ ℝ)
168	167	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (2 · 𝐶) ∈ ℝ)
169	168, 153	rerpdivcld 13130	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((2 · 𝐶) / (√‘𝑥)) ∈ ℝ)
170	169	recnd 11318	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((2 · 𝐶) / (√‘𝑥)) ∈ ℂ)
171	156, 160, 170	adddird 11315	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (((2 · (√‘𝑥)) + (Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) − (2 · (√‘𝑥)))) · ((2 · 𝐶) / (√‘𝑥))) = (((2 · (√‘𝑥)) · ((2 · 𝐶) / (√‘𝑥))) + ((Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) − (2 · (√‘𝑥))) · ((2 · 𝐶) / (√‘𝑥)))))
172	156, 159	pncan3d 11650	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((2 · (√‘𝑥)) + (Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) − (2 · (√‘𝑥)))) = Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)))
173	172	oveq1d 7463	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (((2 · (√‘𝑥)) + (Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) − (2 · (√‘𝑥)))) · ((2 · 𝐶) / (√‘𝑥))) = (Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) · ((2 · 𝐶) / (√‘𝑥))))
174		2cnd 12371	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 2 ∈ ℂ)
175	174, 154, 170	mulassd 11313	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((2 · (√‘𝑥)) · ((2 · 𝐶) / (√‘𝑥))) = (2 · ((√‘𝑥) · ((2 · 𝐶) / (√‘𝑥)))))
176	168	recnd 11318	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (2 · 𝐶) ∈ ℂ)
177	153	rpne0d 13104	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (√‘𝑥) ≠ 0)
178	176, 154, 177	divcan2d 12072	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((√‘𝑥) · ((2 · 𝐶) / (√‘𝑥))) = (2 · 𝐶))
179	178	oveq2d 7464	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (2 · ((√‘𝑥) · ((2 · 𝐶) / (√‘𝑥)))) = (2 · (2 · 𝐶)))
180	175, 179	eqtrd 2780	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((2 · (√‘𝑥)) · ((2 · 𝐶) / (√‘𝑥))) = (2 · (2 · 𝐶)))
181	180	oveq1d 7463	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (((2 · (√‘𝑥)) · ((2 · 𝐶) / (√‘𝑥))) + ((Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) − (2 · (√‘𝑥))) · ((2 · 𝐶) / (√‘𝑥)))) = ((2 · (2 · 𝐶)) + ((Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) − (2 · (√‘𝑥))) · ((2 · 𝐶) / (√‘𝑥)))))
182	171, 173, 181	3eqtr3d 2788	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) · ((2 · 𝐶) / (√‘𝑥))) = ((2 · (2 · 𝐶)) + ((Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) − (2 · (√‘𝑥))) · ((2 · 𝐶) / (√‘𝑥)))))
183	182	mpteq2dva 5266	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) · ((2 · 𝐶) / (√‘𝑥)))) = (𝑥 ∈ ℝ⁺ ↦ ((2 · (2 · 𝐶)) + ((Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) − (2 · (√‘𝑥))) · ((2 · 𝐶) / (√‘𝑥))))))
184		remulcl 11269	. . . . . . . 8 ⊢ ((2 ∈ ℝ ∧ (2 · 𝐶) ∈ ℝ) → (2 · (2 · 𝐶)) ∈ ℝ)
185	161, 167, 184	sylancr 586	. . . . . . 7 ⊢ (𝜑 → (2 · (2 · 𝐶)) ∈ ℝ)
186	185	recnd 11318	. . . . . 6 ⊢ (𝜑 → (2 · (2 · 𝐶)) ∈ ℂ)
187	186	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (2 · (2 · 𝐶)) ∈ ℂ)
188	160, 170	mulcld 11310	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) − (2 · (√‘𝑥))) · ((2 · 𝐶) / (√‘𝑥))) ∈ ℂ)
189		rpssre 13064	. . . . . 6 ⊢ ℝ⁺ ⊆ ℝ
190		o1const 15666	. . . . . 6 ⊢ ((ℝ⁺ ⊆ ℝ ∧ (2 · (2 · 𝐶)) ∈ ℂ) → (𝑥 ∈ ℝ⁺ ↦ (2 · (2 · 𝐶))) ∈ 𝑂(1))
191	189, 186, 190	sylancr 586	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (2 · (2 · 𝐶))) ∈ 𝑂(1))
192		eqid 2740	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) − (2 · (√‘𝑥)))) = (𝑥 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) − (2 · (√‘𝑥))))
193	192	divsqrsum 27043	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) − (2 · (√‘𝑥)))) ∈ dom ⇝_𝑟
194		rlimdmo1 15664	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) − (2 · (√‘𝑥)))) ∈ dom ⇝_𝑟 → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) − (2 · (√‘𝑥)))) ∈ 𝑂(1))
195	193, 194	mp1i 13	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) − (2 · (√‘𝑥)))) ∈ 𝑂(1))
196	176, 154, 177	divrecd 12073	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((2 · 𝐶) / (√‘𝑥)) = ((2 · 𝐶) · (1 / (√‘𝑥))))
197	196	mpteq2dva 5266	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ ((2 · 𝐶) / (√‘𝑥))) = (𝑥 ∈ ℝ⁺ ↦ ((2 · 𝐶) · (1 / (√‘𝑥)))))
198	153	rprecred 13110	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1 / (√‘𝑥)) ∈ ℝ)
199	167	recnd 11318	. . . . . . . . . 10 ⊢ (𝜑 → (2 · 𝐶) ∈ ℂ)
200		rlimconst 15590	. . . . . . . . . 10 ⊢ ((ℝ⁺ ⊆ ℝ ∧ (2 · 𝐶) ∈ ℂ) → (𝑥 ∈ ℝ⁺ ↦ (2 · 𝐶)) ⇝_𝑟 (2 · 𝐶))
201	189, 199, 200	sylancr 586	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (2 · 𝐶)) ⇝_𝑟 (2 · 𝐶))
202		sqrtlim 27034	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ ↦ (1 / (√‘𝑥))) ⇝_𝑟 0
203	202	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (1 / (√‘𝑥))) ⇝_𝑟 0)
204	168, 198, 201, 203	rlimmul 15692	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ ((2 · 𝐶) · (1 / (√‘𝑥)))) ⇝_𝑟 ((2 · 𝐶) · 0))
205	197, 204	eqbrtrd 5188	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ ((2 · 𝐶) / (√‘𝑥))) ⇝_𝑟 ((2 · 𝐶) · 0))
206		rlimo1 15663	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ⁺ ↦ ((2 · 𝐶) / (√‘𝑥))) ⇝_𝑟 ((2 · 𝐶) · 0) → (𝑥 ∈ ℝ⁺ ↦ ((2 · 𝐶) / (√‘𝑥))) ∈ 𝑂(1))
207	205, 206	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ ((2 · 𝐶) / (√‘𝑥))) ∈ 𝑂(1))
208	160, 170, 195, 207	o1mul2 15671	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ ((Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) − (2 · (√‘𝑥))) · ((2 · 𝐶) / (√‘𝑥)))) ∈ 𝑂(1))
209	187, 188, 191, 208	o1add2 15670	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ ((2 · (2 · 𝐶)) + ((Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) − (2 · (√‘𝑥))) · ((2 · 𝐶) / (√‘𝑥))))) ∈ 𝑂(1))
210	183, 209	eqeltrd 2844	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) · ((2 · 𝐶) / (√‘𝑥)))) ∈ 𝑂(1))
211	158, 169	remulcld 11320	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) · ((2 · 𝐶) / (√‘𝑥))) ∈ ℝ)
212	3, 147	fsumcl 15781	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) ∈ ℂ)
213	1, 212	fsumcl 15781	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑑 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) ∈ ℂ)
214	213	abscld 15485	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (abs‘Σ𝑑 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))) ∈ ℝ)
215	211	recnd 11318	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) · ((2 · 𝐶) / (√‘𝑥))) ∈ ℂ)
216	215	abscld 15485	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (abs‘(Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) · ((2 · 𝐶) / (√‘𝑥)))) ∈ ℝ)
217	212	abscld 15485	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))) ∈ ℝ)
218	1, 217	fsumrecl 15782	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑑 ∈ (1...(⌊‘𝑥))(abs‘Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))) ∈ ℝ)
219	1, 212	fsumabs 15849	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (abs‘Σ𝑑 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))) ≤ Σ𝑑 ∈ (1...(⌊‘𝑥))(abs‘Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))))
220	169	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((2 · 𝐶) / (√‘𝑥)) ∈ ℝ)
221	157, 220	remulcld 11320	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((1 / (√‘𝑑)) · ((2 · 𝐶) / (√‘𝑥))) ∈ ℝ)
222	118, 138	syldan 590	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))) → ((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) ∈ ℂ)
223	3, 222	fsumcl 15781	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) ∈ ℂ)
224	223	abscld 15485	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) ∈ ℝ)
225		rpvmasum.a	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℕ)
226		rpvmasum2.1	. . . . . . . . . . 11 ⊢ 1 = (0_g‘𝐺)
227		dchrisum0lem1.f	. . . . . . . . . . 11 ⊢ 𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / (√‘𝑎)))
228		dchrisum0.s	. . . . . . . . . . 11 ⊢ (𝜑 → seq1( + , 𝐹) ⇝ 𝑆)
229		dchrisum0.1	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑆)) ≤ (𝐶 / (√‘𝑦)))
230	120, 122, 225, 119, 121, 226, 123, 125, 227, 162, 228, 229	dchrisum0lem1b 27577	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) ≤ ((2 · 𝐶) / (√‘𝑥)))
231	224, 220, 141, 230	lediv1dd 13157	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((abs‘Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) / (√‘𝑑)) ≤ (((2 · 𝐶) / (√‘𝑥)) / (√‘𝑑)))
232	141	rpcnd 13101	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (√‘𝑑) ∈ ℂ)
233	141	rpne0d 13104	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (√‘𝑑) ≠ 0)
234	223, 232, 233	absdivd 15504	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘(Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))) = ((abs‘Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) / (abs‘(√‘𝑑))))
235	3, 232, 222, 233	fsumdivc 15834	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) = Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)))
236	235	fveq2d 6924	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘(Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))) = (abs‘Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))))
237	141	rprege0d 13106	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((√‘𝑑) ∈ ℝ ∧ 0 ≤ (√‘𝑑)))
238		absid 15345	. . . . . . . . . . . 12 ⊢ (((√‘𝑑) ∈ ℝ ∧ 0 ≤ (√‘𝑑)) → (abs‘(√‘𝑑)) = (√‘𝑑))
239	237, 238	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘(√‘𝑑)) = (√‘𝑑))
240	239	oveq2d 7464	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((abs‘Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) / (abs‘(√‘𝑑))) = ((abs‘Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) / (√‘𝑑)))
241	234, 236, 240	3eqtr3rd 2789	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((abs‘Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) / (√‘𝑑)) = (abs‘Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))))
242	170	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((2 · 𝐶) / (√‘𝑥)) ∈ ℂ)
243	242, 232, 233	divrec2d 12074	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (((2 · 𝐶) / (√‘𝑥)) / (√‘𝑑)) = ((1 / (√‘𝑑)) · ((2 · 𝐶) / (√‘𝑥))))
244	231, 241, 243	3brtr3d 5197	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))) ≤ ((1 / (√‘𝑑)) · ((2 · 𝐶) / (√‘𝑥))))
245	1, 217, 221, 244	fsumle 15847	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑑 ∈ (1...(⌊‘𝑥))(abs‘Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))) ≤ Σ𝑑 ∈ (1...(⌊‘𝑥))((1 / (√‘𝑑)) · ((2 · 𝐶) / (√‘𝑥))))
246	157	recnd 11318	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (1 / (√‘𝑑)) ∈ ℂ)
247	1, 170, 246	fsummulc1 15833	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) · ((2 · 𝐶) / (√‘𝑥))) = Σ𝑑 ∈ (1...(⌊‘𝑥))((1 / (√‘𝑑)) · ((2 · 𝐶) / (√‘𝑥))))
248	245, 247	breqtrrd 5194	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑑 ∈ (1...(⌊‘𝑥))(abs‘Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))) ≤ (Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) · ((2 · 𝐶) / (√‘𝑥))))
249	214, 218, 211, 219, 248	letrd 11447	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (abs‘Σ𝑑 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))) ≤ (Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) · ((2 · 𝐶) / (√‘𝑥))))
250	211	leabsd 15463	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) · ((2 · 𝐶) / (√‘𝑥))) ≤ (abs‘(Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) · ((2 · 𝐶) / (√‘𝑥)))))
251	214, 211, 216, 249, 250	letrd 11447	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (abs‘Σ𝑑 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))) ≤ (abs‘(Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) · ((2 · 𝐶) / (√‘𝑥)))))
252	251	adantrr 716	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘Σ𝑑 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))) ≤ (abs‘(Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) · ((2 · 𝐶) / (√‘𝑥)))))
253	151, 210, 211, 213, 252	o1le 15701	. 2 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ Σ𝑑 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))) ∈ 𝑂(1))
254	150, 253	eqeltrrd 2845	1 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))) ∈ 𝑂(1))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 ∀wral 3067 {crab 3443 ∖ cdif 3973 ∪ cun 3974 ⊆ wss 3976 {csn 4648 class class class wbr 5166 ↦ cmpt 5249 dom cdm 5700 ‘cfv 6573 (class class class)co 7448 ℂcc 11182 ℝcr 11183 0cc0 11184 1c1 11185 + caddc 11187 · cmul 11189 +∞cpnf 11321 < clt 11324 ≤ cle 11325 − cmin 11520 / cdiv 11947 ℕcn 12293 2c2 12348 ℕ₀cn0 12553 ℤcz 12639 ℤ_≥cuz 12903 ℝ⁺crp 13057 [,)cico 13409 ...cfz 13567 ⌊cfl 13841 seqcseq 14052 ↑cexp 14112 √csqrt 15282 abscabs 15283 ⇝ cli 15530 ⇝_𝑟 crli 15531 𝑂(1)co1 15532 Σcsu 15734 Basecbs 17258 0_gc0g 17499 ℤRHomczrh 21533 ℤ/nℤczn 21536 DChrcdchr 27294

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263 ax-mulf 11264

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-tpos 8267 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-ec 8765 df-qs 8769 df-map 8886 df-pm 8887 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-fi 9480 df-sup 9511 df-inf 9512 df-oi 9579 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-q 13014 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-ioo 13411 df-ioc 13412 df-ico 13413 df-icc 13414 df-fz 13568 df-fzo 13712 df-fl 13843 df-mod 13921 df-seq 14053 df-exp 14113 df-fac 14323 df-bc 14352 df-hash 14380 df-shft 15116 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-limsup 15517 df-clim 15534 df-rlim 15535 df-o1 15536 df-lo1 15537 df-sum 15735 df-ef 16115 df-sin 16117 df-cos 16118 df-pi 16120 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-rest 17482 df-topn 17483 df-0g 17501 df-gsum 17502 df-topgen 17503 df-pt 17504 df-prds 17507 df-xrs 17562 df-qtop 17567 df-imas 17568 df-qus 17569 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-mhm 18818 df-submnd 18819 df-grp 18976 df-minusg 18977 df-sbg 18978 df-mulg 19108 df-subg 19163 df-nsg 19164 df-eqg 19165 df-ghm 19253 df-cntz 19357 df-cmn 19824 df-abl 19825 df-mgp 20162 df-rng 20180 df-ur 20209 df-ring 20262 df-cring 20263 df-oppr 20360 df-dvdsr 20383 df-unit 20384 df-rhm 20498 df-subrng 20572 df-subrg 20597 df-lmod 20882 df-lss 20953 df-lsp 20993 df-sra 21195 df-rgmod 21196 df-lidl 21241 df-rsp 21242 df-2idl 21283 df-psmet 21379 df-xmet 21380 df-met 21381 df-bl 21382 df-mopn 21383 df-fbas 21384 df-fg 21385 df-cnfld 21388 df-zring 21481 df-zrh 21537 df-zn 21540 df-top 22921 df-topon 22938 df-topsp 22960 df-bases 22974 df-cld 23048 df-ntr 23049 df-cls 23050 df-nei 23127 df-lp 23165 df-perf 23166 df-cn 23256 df-cnp 23257 df-haus 23344 df-cmp 23416 df-tx 23591 df-hmeo 23784 df-fil 23875 df-fm 23967 df-flim 23968 df-flf 23969 df-xms 24351 df-ms 24352 df-tms 24353 df-cncf 24923 df-limc 25921 df-dv 25922 df-log 26616 df-cxp 26617 df-dchr 27295

This theorem is referenced by: dchrisum0lem3 27581

W3C validator